When Saudi Arabia’s Crown Prince Mohammed bin Salman toured the U.S. two years ago, he couldn’t even get an audience with Tesla Inc. Chief Executive Elon Musk, people familiar with the outreach said. Since then, his country’s sovereign-wealth fund has boosted its stake in the car maker, and he is now weighing whether to be part of a deal to take the company private.

The shift shows how quickly Crown Prince Mohammed and his sovereign-wealth fund have become a pivotal global investor. The Public Investment Fund’s moves have...

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